As a tip, when you are looking for usage on Google, add quotes. It will ask Google to treat your search more or less literally instead of trying a whole bunch of linguistic parsing (like including synonyms, spelling variations, topic parsing, etc.):
"Center of Excellence" - 47.8 million results
"Center of Competence" - 356 thousand results
Results are similar on Bing.
This should give you a pretty good idea of the popularity of both expressions.
If you want to delve deeper into the popularity in US English, you can restrict your search to ".us" websites:
"Center of Excellence" site:*.us - 42.4 thousand results
"Center of Competence" site:*.us - 8.6 thousand results
"Center of Excellence" site:*.us - 46.1 thousand results
"Center of Competence" site:*.uk - 14.7 thousand results
It should be clear that "Center of Excellence" is the more common phrase but that "Center of competence" is commonly used as well.
And yes, of course, Google result count is not perfect, but it does give a reasonably objective measure of the popularity of various terms.